Posted on Categories:Generalized linear model, 广义线性模型, 统计代写, 统计代考

# 统计代写|广义线性模型代写Generalized linear model代考|PHC7098 Partially Observed Information

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 统计代写|广义线性模型代写Generalized linear model代考|Partially Observed Information

One important assumption that we have made in the application of EMM is (2.14). This assumption holds, for example, if the random effects and errors are symmetrically distributed. However, from a practical point of view, such an assumption is not very pleasant because, like normality, symmetry may not hold in practice. Furthermore, it has been empirically observed that estimators of the higher (e.g., fourth) moment of the random effects is unstable. An alternative method, known as partially observed information (POI), was proposed in Sect. 1.4.2 for estimating the asymptotic covariance matrices of REML or ML estimators. The method applies to a general non-Gaussian mixed ANOVA model regardless of (2.14). Let us consider the same example above but apply the POI method this time.
Example 2.2 (Continued) Suppose that one wishes to test the hypothesis $\mathrm{H}0: \gamma_1=$ 1 ; that is, the variance contribution due to the random effects is the same as that due to the errors. Note that in this case $\theta=\left(\lambda, \gamma_1\right)^{\prime}$, so the null hypothesis corresponds to (2.8) with $K=(0,1)^{\prime}$ and $c=1$. Furthermore, we have $K^{\prime} \Sigma{\mathrm{R}} K=\Sigma_{\mathrm{R}, 11}$, which is the asymptotic variance of $\hat{\gamma}1$, the REML estimator of $\gamma_1$. Thus, the test statistic is $\hat{\chi}^2=\left(\hat{\gamma}_1-1\right)^2 / \hat{\Sigma}{\mathrm{R}, 11}$, where $\hat{\Sigma}{\mathrm{R}, 11}$ is the POQUIM estimator of $\Sigma{\mathrm{R}, 11}$ (see Sect. 1.8.5) given by

$$\hat{\Sigma}{\mathrm{R}, 11}=\frac{\hat{\mathcal{I}}{1,11} \hat{\mathcal{I}}{2,00}^2-2 \hat{\mathcal{I}}{1,01} \hat{\mathcal{I}}{2,00} \hat{\mathcal{I}}{2,01}+\hat{\mathcal{I}}{1,00} \hat{\mathcal{I}}{2,01}^2}{\left(\hat{\mathcal{I}}{2,00} \hat{\mathcal{I}}{2,11}-\hat{\mathcal{I}}{2,01}^2\right)^2}$$ where $\hat{\mathcal{I}}{1, s t}=\hat{\mathcal{I}}{1,1, s t}+\hat{\mathcal{I}}{1,2, s t}, s, t=0,1$, and $\hat{\mathcal{I}}{1, r, s t}, r=1,2$ are given in Example 1.1 (Continued) in Sect. 1.8.5 but with $\hat{\gamma}_1$ replaced by 1 , its value under $\mathrm{H}_0$; furthermore, we have $$\hat{\mathcal{I}}{2,00}=-\frac{(m k-1)}{2 \hat{\lambda}^2}, \hat{\mathcal{I}}{2,01}=-\frac{(m-1) k}{2 \hat{\lambda}^{\left(1+\hat{\gamma}_1 k\right)}}, \hat{\mathcal{I}}{2,11}=-\frac{(m-1) k^2}{2\left(1+\hat{\gamma}_1 k\right)^2},$$
again with $\hat{\gamma}_1$ replaced by 1 , where $\hat{\lambda}$ is the REML estimator of $\lambda$ (Exercise 2.8). The asymptotic null distribution of the test is $\chi_1^2$.

## 统计代写|广义线性模型代写Generalized linear model代考|Jackknife Method

For a non-Gaussian longitudinal LMM, the asymptotic covariance matrix of the REML (ML) estimator may also be estimated using the jackknife method discussed in Sect. 1.4.4. One advantage of the jackknife method is that it is one-formula-forall. In fact, the same jackknife estimator not only applies to longitudinal LMM, it also applies to longitudinal generalized linear mixed models, which we discuss later in Chaps. 3 and 4 . Let $\psi$ be the vector of all the parameters involved in a nonGaussian longitudinal LMM, which includes fixed effects and variance components. Let $\hat{\psi}$ be the REML or ML estimator of $\psi$. Then, the jackknife estimator of the asymptotic covariance matrix of $\hat{\psi}$ is given by (1.43). Jiang and Lahiri (2004) showed that, under suitable conditions, the jackknife estimator is consistent in the sense that $\hat{\Sigma}{\text {Jack }}=\Sigma+O{\mathrm{P}}\left(m^{-1-\delta}\right)$ for some $\delta>0$. Note that, typically, we have $\Sigma=O\left(m^{-1}\right)$. Therefore, one may use $\hat{\Sigma}=\hat{\Sigma}_{\text {Jack }}$ on the left side of (2.10) for the asymptotic test. We illustrate with a simple example.

Example 2.4 (The James-Stein estimator) Let $y_i, i=1, \ldots, m$ be independent such that $y_i \sim N\left(\theta_i, 1\right)$. In the context of simultaneous estimation of $\theta=$ $\left(\theta_1, \ldots, \theta_m\right)^{\prime}$, it is well-known that for $m \geq 3$, the James-Stein estimator dominates the maximum likelihood estimator, given by $y=\left(y_1, \ldots, y_m\right)^{\prime}$, in terms of the frequentist risk under the sum of squared error loss function (e.g., Lehmann and Casella 1998, pp. 272-273). Efron and Morris (1973) provided an empirical Bayes justification of the James-Stein estimator. Their Bayesian model can be equivalently written as the following simple random effects model: $y_i=\alpha_i+\epsilon_i, i=1, \ldots, m$, where the sampling errors $\left{\epsilon_i\right}$ and the random effects $\left{\alpha_i\right}$ are, respectively, independently distributed with $\alpha_i \sim N(0, \psi)$ and $\epsilon_i \sim N(0,1)$ and $\epsilon$ and $\alpha$ are independent.

## 统计代写|广义线性模型代写Generalized linear model代考|Partially Observed Information

$$\hat{\Sigma} \mathrm{R}, 11=\frac{\hat{\mathcal{I}} 1,11 \hat{\mathcal{I}} 2,00^2-2 \hat{\mathcal{I}} 1,01 \hat{\mathcal{I}} 2,00 \hat{\mathcal{I}} 2,01+\hat{\mathcal{I}} 1,00 \hat{\mathcal{I}} 2,01^2}{\left(\hat{\mathcal{I}} 2,00 \hat{\mathcal{I}} 2,11-\hat{\mathcal{I}} 2,01^2\right)^2}$$

$$\hat{\mathcal{I}} 2,00=-\frac{(m k-1)}{2 \hat{\lambda}^2}, \hat{\mathcal{I}} 2,01=-\frac{(m-1) k}{2 \hat{\lambda}^{\left(1+\hat{\gamma}_1 k\right)}}, \hat{\mathcal{I}} 2,11=-\frac{(m-1) k^2}{2\left(1+\hat{\gamma}_1 k\right)^2},$$

## 统计代写|广义线性模妍代写Generalized linear model代写|Jackknife Method

$y_i=\alpha_i+\epsilon_i, i=1, \ldots, m$ ，其中抽样误差 $\backslash$ left 缺少或无法识别的分隔符

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。